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   "source": [
    "## 循环神经网络"
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   "source": [
    "### 序列模型\n",
    "#### 引言\n",
    "* 锚定（anchoring）效应：基于其他人的意见做出评价。\n",
    "* 享乐适应（hedonic adaption）：人们迅速接受并且适应一种更好或者更坏的情况 作为新的常态。\n",
    "* 季节性（seasonality）：少有观众喜欢在八月看圣诞老人的电影。\n",
    "* 有时，电影会由于导演或演员在制作中的不当行为变得不受欢迎。\n",
    "* 有些电影因为其极度糟糕只能成为小众电影。\n",
    "* 在使用应用程序时，许多用户都有很强的特定习惯。\n",
    "* 预测明天的股价要比过去的股价更困难，尽管两者都只是估计一个数字。 毕竟，先见之明比事后诸葛亮难得多。\n",
    "* 在本质上，音乐、语音、文本和视频都是连续的。 如果它们的序列被我们重排，那么就会失去原有的意义。 \n",
    "* 事实上，地震是时空相关的，即余震通常发生在很短的时间跨度和很近的距离内。\n",
    "* 人类之间的互动也是连续的，这可以从微博上的争吵和辩论中看出。"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 统计工具"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![ftse100.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h0pkazoeg1j30rk0g2tcd.jpg)      \n",
    "其中，用$x_{t}$表示价格，即在时间步（time step）$t\\in Z^{+}$时，观察到的价格$x_{t}$。 请注意，t对于本文中的序列通常是离散的，并在整数或其子集上变化。 假设一个交易员想在t日的股市中表现良好，于是通过以下途径预测$x_{t}$：\n",
    "$$x_{t} \\sim P(x_{t}|x_{t-1},x_{t-2},...,x_{1})$$"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "自回归模型\n",
    "* 输入数据的数量这个数字将会随着我们遇到的数据量的增加而增加， 因此需要一个近似方法来使这个计算变得容易处理。\n",
    "* 第一种策略，假设在现实情况下相当长的序列$x_{t-1},...,x_{1}$可能是不必要的， 因此我们只需要满足某个长度为T的时间跨度， 即使用观测序列$x_{t-1},...,x_{t-T}$。 当下获得的最直接的好处就是参数的数量总是不变的， 至少t>T在时如此，这就使我们能够训练一个上面提及的深度网络。（自回归模型（autoregressive models）， 因为它们是对自己执行回归）\n",
    "* 第二种策略，如 下图所示， 是保留一些对过去观测的总结$h_{t}$， 并且同时更新预测$\\hat{x}_{t}$和总结$h_{t}$。 这就产生了基于$\\hat{x}_{t} = P(x_{t}|h_{t})$估计$x_{t}$， 以及公式$h_{t} = g(h_{t-1},x_{t-1})$更新的模型。 由于$h_{t}$.从未被观测到，这类模型也被称为 隐变量自回归模型.        \n",
    "![隐变量回归.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h0pl51ofjvj30b206g3z9.jpg)      \n",
    "* 如何生成训练数据？ 一个经典方法是使用历史观测来预测下一个未来观测。\n",
    "* 整个序列的估计值都将通过以下的方式获得：      \n",
    "![公式.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h0pl6y2jh2j30a001iq31.jpg)     "
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "马尔可夫模型"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 如果T= 1，得到一个 一阶马尔可夫模型（first-order Markov model）， 由P(x)下式给出：        \n",
    "![P(x).png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h0pwxpogalj30c801j74k.jpg)     \n",
    "* 当假设$x_{t}$仅是离散值时，这样的模型特别棒， 因为在这种情况下，使用动态规划可以沿着马尔可夫链精确地计算结果。      \n",
    "![马尔可夫.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h0pwz3hbemj30by04rwff.jpg)     \n",
    "* 很明显，未来的事件不能影响过去。如果我们改变$x_{t}$，基于过去事件得到的分布不会改变。 因此，解释$P(x_{t+1}|x_{t})$应该比解释$P(x_{t}|x_{t+1})$\n",
    "更容易。"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 训练\n",
    "* 首先，我们生成一些数据：使用正弦函数和一些可加性噪声来生成序列数据， 时间步为1，2，...，1000。"
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    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import torch\n",
    "from torch import nn\n",
    "from d2l import torch as d2l\n",
    "\n",
    "T = 1000 # 产生1000个点\n",
    "time = torch.arange(1,T+1,dtype=torch.float32)\n",
    "x = torch.sin(0.01*time) + torch.normal(0,0.2,(T,))\n",
    "d2l.plot(time,[x],'time','x',xlim=[1,1000],figsize=(6,3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 接下来，我们将这个序列转换为模型的“特征－标签”（feature-label）对。\n",
    "* 如果没有足够的历史记录来描述前T个数据样本；如果拥有足够长的序列就丢弃这几项； 另一个方法是用零填充序列。在这里，我们仅使用前600个“特征－标签”对进行训练。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "tau = 4\n",
    "features = torch.zeros((T - tau,tau))\n",
    "for i in range(tau):\n",
    "    features[:,i] = x[i:T-tau+i]\n",
    "labels = x[tau:].reshape((-1,1))\n",
    "batch_size,n_train = 16,600\n",
    "# 只有前n个样本用于训练\n",
    "train_iter = d2l.load_array((features[:n_train],labels[:n_train]),\n",
    "                            batch_size,is_train=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "使用一个相当简单的架构训练模型： 一个拥有两个全连接层的多层感知机，ReLU激活函数和平方损失。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 初始化网络权重的函数\n",
    "def init_weights(m):\n",
    "    if type(m) == nn.Linear:\n",
    "        nn.init.xavier_uniform_(m.weight)\n",
    "\n",
    "# 一个简单的多层感知机\n",
    "def get_net():\n",
    "    net = nn.Sequential(nn.Linear(4, 10),\n",
    "                        nn.ReLU(),\n",
    "                        nn.Linear(10, 1))\n",
    "    net.apply(init_weights)\n",
    "    return net\n",
    "\n",
    "# 平方损失。注意：MSELoss计算平方误差时不带系数1/2\n",
    "loss = nn.MSELoss(reduction='none')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "准备训练模型"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 1, loss: 0.062637\n",
      "epoch 2, loss: 0.058825\n",
      "epoch 3, loss: 0.058700\n",
      "epoch 4, loss: 0.054355\n",
      "epoch 5, loss: 0.052925\n"
     ]
    }
   ],
   "source": [
    "def train(net, train_iter, loss, epochs, lr):\n",
    "    trainer = torch.optim.Adam(net.parameters(), lr)\n",
    "    for epoch in range(epochs):\n",
    "        for X, y in train_iter:\n",
    "            trainer.zero_grad()\n",
    "            l = loss(net(X), y)\n",
    "            l.sum().backward()\n",
    "            trainer.step()\n",
    "        print(f'epoch {epoch + 1}, '\n",
    "              f'loss: {d2l.evaluate_loss(net, train_iter, loss):f}')\n",
    "\n",
    "net = get_net()\n",
    "train(net, train_iter, loss, 5, 0.01)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 预测\n",
    "首先是检查模型预测下一个时间步的能力， 也就是单步预测（one-step-ahead prediction）。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
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   "source": [
    "onestep_preds = net(features)\n",
    "d2l.plot([time,time[tau:]],\n",
    "        [x.detach().numpy(),onestep_preds.detach().numpy()],\n",
    "        'time','x',\n",
    "        legend=['data','1-step preds'],\n",
    "        xlim=[1,1000],figsize=(6,3))"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "对于直到$x_{t}$的观测序列，其在时间步t+k处的预测输出$\\hat x_{t+k}$\n",
    " 称为k步预测（k-step-ahead-prediction）。 由于我们的观察已经到了$x_{604}$，它的步预测是$\\hat x_{604+k}$。换句话说，我们必须使用我们自己的预测（而不是原始数据）来进行多步预测。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
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     },
     "metadata": {
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   ],
   "source": [
    "multistep_preds = torch.zeros(T)\n",
    "multistep_preds[:n_train+tau] = x[: n_train+tau]\n",
    "for i in range(n_train+tau,T):\n",
    "    multistep_preds[i-tau:i].reshape((1,-1))\n",
    "\n",
    "d2l.plot([time,time[tau:],time[n_train+tau:]],\n",
    "        [x.detach().numpy(),onestep_preds.detach().numpy(),\n",
    "        multistep_preds[n_train+tau:].detach().numpy()],\n",
    "        'time','x',\n",
    "        legend=['data','1-step preds','multistep preds'],\n",
    "        xlim=[1,1000],figsize=(6,3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 为什么这个算法效果这么差呢？事实是由于错误的累积      \n",
    "![out.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h0qo98ty6kj30gu02sgo6.jpg)      \n",
    "* 基于k=1,4,16,64，通过对整个序列预测的计算， 让我们更仔细地看一下k步预测的困难。"
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      ],
      "text/plain": [
       "<Figure size 432x216 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "max_steps = 64\n",
    "features = torch.zeros(T-tau-max_steps+1,tau+max_steps)\n",
    "# 列i（i<tau）是来自x的观测，其时间步从（i+1）到（i+T-tau-max_steps+1）\n",
    "for i in range(tau):\n",
    "    features[:,i] = x[i:i+T-tau-max_steps+1]\n",
    "\n",
    "# 列i（i>=tau）是来自（i-tau+1）步的预测，其时间步从（i+1）到（i+T-tau-max_steps+1）\n",
    "for i in range(tau,tau+max_steps):\n",
    "    features[:,i] = net(features[:,i-tau:i]).reshape(-1)\n",
    "\n",
    "steps = [1,4,16,64]\n",
    "d2l.plot([time[tau+i-1:T-max_steps+i] for i in steps],\n",
    "        [features[:,(tau+i-1)].detach().numpy() for i in steps],\n",
    "        'time','x',\n",
    "        legend=[f'{i}-step preds' for i in steps],\n",
    "        xlim=[5,1000],figsize=(6,3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 小结\n",
    "* 内插法（在现有观测值之间进行估计）和外推法（对超出已知观测范围进行预测）在实践的难度上差别很大。因此，对于你所拥有的序列数据，在训练时始终要尊重其时间顺序，即最好不要基于未来的数据进行训练。\n",
    "* 序列模型的估计需要专门的统计工具，两种较流行的选择是自回归模型和隐变量自回归模型。\n",
    "* 对于时间是向前推进的因果模型，正向估计通常比反向估计更容易。\n",
    "* 对于直到时间步t的观测序列，其在时间步t+k的预测输出是“k步预测”。随着我们对预测时间k值的增加，会造成误差的快速累积和预测质量的极速下降。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 文本预处理"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "解析文本的常见预处理步骤：\n",
    "* 1.将文本作为字符串加载到内存中。\n",
    "* 2.将字符串拆分为词元（如单词和字符）。\n",
    "* 3.建立一个词表，将拆分的词元映射到数字索引。\n",
    "* 4.将文本转换为数字索引序列，方便模型操作。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "import collections\n",
    "import re\n",
    "from d2l import torch as d2l"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 读取数据集  \n",
    "* 下面的函数将数据集读取到由多条文本行组成的列表中，其中每条文本行都是一个字符串。 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "# 文本总行数：3221\n",
      "the time machine by h g wells\n",
      "fire burned brightly and the soft radiance of the incandescent\n"
     ]
    }
   ],
   "source": [
    "d2l.DATA_HUB['time_machine'] = (d2l.DATA_URL + 'timemachine.txt','090b5e7e70c295757f55df93cb0a180b9691891a')\n",
    "\n",
    "def read_time_machine():\n",
    "    \"\"\"将时间机器数据集加载到文本行的列表中\"\"\"\n",
    "    with open(d2l.download('time_machine'),'r') as f:\n",
    "        lines = f.readlines()\n",
    "    return [re.sub('[^A-Za-z]+',' ',line).strip().lower() for line in lines]\n",
    "\n",
    "lines = read_time_machine()\n",
    "print(f'# 文本总行数：{len(lines)}')\n",
    "print(lines[0])\n",
    "print(lines[11])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 词元化\n",
    "* tokenize函数将文本行列表（lines）作为输入， 列表中的每个元素是一个文本序列（如一条文本行）。 \n",
    "* 每个文本序列又被拆分成一个词元列表，词元（token）是文本的基本单位。 \n",
    "* 最后，返回一个由词元列表组成的列表，其中的每个词元都是一个字符串（string）。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['the', 'time', 'machine', 'by', 'h', 'g', 'wells']\n",
      "[]\n",
      "[]\n",
      "[]\n",
      "[]\n",
      "['i']\n",
      "[]\n",
      "[]\n",
      "['the', 'time', 'traveller', 'for', 'so', 'it', 'will', 'be', 'convenient', 'to', 'speak', 'of', 'him']\n",
      "['was', 'expounding', 'a', 'recondite', 'matter', 'to', 'us', 'his', 'grey', 'eyes', 'shone', 'and']\n",
      "['twinkled', 'and', 'his', 'usually', 'pale', 'face', 'was', 'flushed', 'and', 'animated', 'the']\n"
     ]
    }
   ],
   "source": [
    "def tokenize(lines,token='word'):\n",
    "    \"\"\"将文本行拆分为单词或字符词元\"\"\"\n",
    "    if token == 'word':\n",
    "        return [line.split() for line in lines]\n",
    "    elif token == 'char':\n",
    "        return [list(line) for line in lines]\n",
    "    else:\n",
    "        print('错误：未知词元类型：'+token)\n",
    "\n",
    "tokens = tokenize(lines)\n",
    "for i in range(11):\n",
    "    print(tokens[i])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 词表\n",
    "* 词元的类型是字符串，而模型需要的输入是数字，因此这种类型不方便模型使用。\n",
    "* 让我们构建一个字典，通常也叫做词表（vocabulary）， 用来将字符串类型的词元映射到从0开始的数字索引中。\n",
    "* 我们先将训练集中的所有文档合并在一起，对它们的唯一词元进行统计， 得到的统计结果称之为语料（corpus）。\n",
    "* 然后根据每个唯一词元的出现频率，为其分配一个数字索引。 很少出现的词元通常被移除，这可以降低复杂性。 \n",
    "* 语料库中不存在或已删除的任何词元都将映射到一个特定的未知词元“< unk>”。 我们可以选择增加一个列表，用于保存那些被保留的词元， 例如：填充词元（“< pad>”）；"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Vocab:  #@save\n",
    "    \"\"\"文本词表\"\"\"\n",
    "    def __init__(self, tokens=None, min_freq=0, reserved_tokens=None):\n",
    "        if tokens is None:\n",
    "            tokens = []\n",
    "        if reserved_tokens is None:\n",
    "            reserved_tokens = []\n",
    "        # 按出现频率排序\n",
    "        counter = count_corpus(tokens)\n",
    "        self._token_freqs = sorted(counter.items(), key=lambda x: x[1],\n",
    "                                   reverse=True)\n",
    "        # 未知词元的索引为0\n",
    "        self.idx_to_token = ['<unk>'] + reserved_tokens\n",
    "        self.token_to_idx = {token: idx\n",
    "                             for idx, token in enumerate(self.idx_to_token)}\n",
    "        for token, freq in self._token_freqs:\n",
    "            if freq < min_freq:\n",
    "                break\n",
    "            if token not in self.token_to_idx:\n",
    "                self.idx_to_token.append(token)\n",
    "                self.token_to_idx[token] = len(self.idx_to_token) - 1\n",
    "\n",
    "    def __len__(self):\n",
    "        return len(self.idx_to_token)\n",
    "\n",
    "    def __getitem__(self, tokens):\n",
    "        if not isinstance(tokens, (list, tuple)):\n",
    "            return self.token_to_idx.get(tokens, self.unk)\n",
    "        return [self.__getitem__(token) for token in tokens]\n",
    "\n",
    "    def to_tokens(self, indices):\n",
    "        if not isinstance(indices, (list, tuple)):\n",
    "            return self.idx_to_token[indices]\n",
    "        return [self.idx_to_token[index] for index in indices]\n",
    "\n",
    "    @property\n",
    "    def unk(self):  # 未知词元的索引为0\n",
    "        return 0\n",
    "\n",
    "    @property\n",
    "    def token_freqs(self):\n",
    "        return self._token_freqs\n",
    "\n",
    "def count_corpus(tokens):  #@save\n",
    "    \"\"\"统计词元的频率\"\"\"\n",
    "    # 这里的tokens是1D列表或2D列表\n",
    "    if len(tokens) == 0 or isinstance(tokens[0], list):\n",
    "        # 将词元列表展平成一个列表\n",
    "        tokens = [token for line in tokens for token in line]\n",
    "    return collections.Counter(tokens)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "首先使用时光机器数据集作为语料库来构建词表，然后打印前几个高频词元及其索引。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[('<unk>', 0), ('the', 1), ('i', 2), ('and', 3), ('of', 4), ('a', 5), ('to', 6), ('was', 7), ('in', 8), ('that', 9), ('my', 10), ('it', 11), ('had', 12), ('me', 13), ('as', 14)]\n"
     ]
    }
   ],
   "source": [
    "vocab = Vocab(tokens)\n",
    "print(list(vocab.token_to_idx.items())[:15])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们可以将每一条文本行转换成一个数字索引列表"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "文本： ['the', 'time', 'machine', 'by', 'h', 'g', 'wells']\n",
      "索引： [1, 19, 50, 40, 2183, 2184, 400]\n",
      "文本： ['twinkled', 'and', 'his', 'usually', 'pale', 'face', 'was', 'flushed', 'and', 'animated', 'the']\n",
      "索引： [2186, 3, 25, 1044, 362, 113, 7, 1421, 3, 1045, 1]\n",
      "文本： ['lights', 'in', 'the', 'lilies', 'of', 'silver', 'caught', 'the', 'bubbles', 'that', 'flashed', 'and']\n",
      "索引： [2189, 8, 1, 2190, 4, 589, 363, 1, 2191, 9, 826, 3]\n"
     ]
    }
   ],
   "source": [
    "for i in [0,10,12]:\n",
    "    print('文本：',tokens[i])\n",
    "    print('索引：',vocab[tokens[i]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 整合所有功能\n",
    "* 将所有功能打包到load_corpus_time_machine函数中， 该函数返回corpus（词元索引列表）和vocab（时光机器语料库的词表）。\n",
    "* 时光机器数据集中的每个文本行不一定是一个句子或一个段落，还可能是一个单词，因此返回的corpus仅处理为单个列表，而不是使用多词元列表构成的一个列表。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(170580, 28)"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def load_corpus_time_machine(max_tokens=-1):\n",
    "    \"\"\"返回时间机器数据集的词元索引列表和词表\"\"\"\n",
    "    lines = read_time_machine()\n",
    "    tokens = tokenize(lines,'char')\n",
    "    vocab = Vocab(tokens)\n",
    "    # 因为时间机器数据集中的每个文本行不一定是一个句子或一个段落\n",
    "    # 所以将所有文本行展平到一个列表中\n",
    "    corpus = [vocab[tokens] for line in tokens for tokens in line]\n",
    "    if max_tokens > 0:\n",
    "        corpus = corpus[:max_tokens]\n",
    "    return corpus,vocab\n",
    "\n",
    "corpus,vocab = load_corpus_time_machine()\n",
    "len(corpus),len(vocab)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 小结 \n",
    "* 文本是序列数据的一种最常见的形式之一。\n",
    "* 为了对文本进行预处理，我们通常将文本拆分为词元，构建词表将词元字符串映射为数字索引，并将文本数据转换为词元索引以供模型操作。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 语言模型和数据集"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 在给定这样的文本序列时，语言模型（language model）的目标是估计序列的联合概率$P(x_{1},x_{2},...,x_{T})$\n",
    "* 只需要基于前面的对话片断中的文本， 就足以生成一个有意义的对话。 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 学习语言模型\n",
    "* 基本概率规则:\n",
    "    $$P(x_{1},x_{2},...,x_{T}) = \\prod_{t=1}^{T}P(x_{t}|x_{1},...,x_{t-1})$$\n",
    "* 训练数据集中词的概率可以根据给定词的相对词频来计算。 例如，可以将估计值$\\hat{P}(deep)$计算为任何以单词“deep”开头的句子的概率。\n",
    "* 一种常见的策略是执行某种形式的拉普拉斯平滑（Laplace smoothing）， 具体方法是在所有计数中添加一个小常量。 用n表示训练集中的单词总数，用m表示唯一单词的数量。       \n",
    "![two.png](https://tva1.sinaimg.cn/large/005T39qaly1h0zzcxyau9j30ca04vq3w.jpg)\n",
    "* 然而，这样的模型很容易变得无效，原因如下： 首先，我们需要存储所有的计数； 其次，这完全忽略了单词的意思。 例如，“猫”（cat）和“猫科动物”（feline）可能出现在相关的上下文中， 但是想根据上下文调整这类模型其实是相当困难的。\n",
    "* 最后，长单词序列大部分是没出现过的， 因此一个模型如果只是简单地统计先前“看到”的单词序列频率， 那么模型面对这种问题肯定是表现不佳的。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 马尔可夫模型与n元语法\n",
    "* 如果$P(x_{t+1}|x_{t},...,x_{1}) = P(x_{t+1}|x_{t})$， 则序列上的分布满足一阶马尔可夫性质。 阶数越高，对应的依赖关系就越长。       \n",
    "![two.png](https://tva1.sinaimg.cn/large/005T39qaly1h0yxlt4py0j30em029glz.jpg)      \n",
    "* 涉及一个、两个和三个变量的概率公式分别被称为 “一元语法”（unigram）、“二元语法”（bigram）和“三元语法”（trigram）模型。 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 自然语言统计\n",
    "时光机器数据集构建词表， 并打印前10个最常用的（频率最高的）单词"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('the', 2261),\n",
       " ('i', 1267),\n",
       " ('and', 1245),\n",
       " ('of', 1155),\n",
       " ('a', 816),\n",
       " ('to', 695),\n",
       " ('was', 552),\n",
       " ('in', 541),\n",
       " ('that', 443),\n",
       " ('my', 440)]"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import random\n",
    "import torch \n",
    "from d2l import torch as d2l\n",
    "\n",
    "tokens = d2l.tokenize(d2l.read_time_machine())\n",
    "# 因为每个文本行不一定是一个句子或段落，因此我们把所有文本行拼接到一起\n",
    "corpus = [token for line in tokens for token in line]\n",
    "vocab = d2l.Vocab(corpus)\n",
    "vocab.token_freqs[:10]"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "最流行的词看起来很无聊， 这些词通常被称为停用词（stop words），因此可以被过滤掉。 尽管如此，它们本身仍然是有意义的，我们仍然会在模型中使用它们。 此外，还有个明显的问题是词频衰减的速度相当地快。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
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   ],
   "source": [
    "freqs = [freq for token,freq in vocab.token_freqs]\n",
    "d2l.plot(freqs,xlabel='token:x',ylabel='frequency:n(x)',xscale='log',yscale='log')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 词频以一种明确的方式迅速衰减。 将前几个单词作为例外消除后，剩余的所有单词大致遵循双对数坐标图上的一条直线。 这意味着单词的频率满足齐普夫定律（Zipf’s law）， 即第i个最常用单词的频率$n_{i}$为：\n",
    "$$logn_{i} =  -\\alpha logi + c$$    \n",
    "* 其中$ \\alpha$是刻画分布的指数，c是常数。         \n",
    "* 这告诉我们想要通过计数统计和平滑来建模单词是不可行的， 因为这样建模的结果会大大高估尾部单词的频率，也就是所谓的不常用单词。 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "看看二元语法的频率是否与一元语法的频率表现出相同的行为方式"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(('of', 'the'), 309),\n",
       " (('in', 'the'), 169),\n",
       " (('i', 'had'), 130),\n",
       " (('i', 'was'), 112),\n",
       " (('and', 'the'), 109),\n",
       " (('the', 'time'), 102),\n",
       " (('it', 'was'), 99),\n",
       " (('to', 'the'), 85),\n",
       " (('as', 'i'), 78),\n",
       " (('of', 'a'), 73)]"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bigram_tokens = [pair for pair in zip(corpus[:-1],corpus[1:])]\n",
    "bigram_vocab = d2l.Vocab(bigram_tokens)\n",
    "bigram_vocab.token_freqs[:10]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 在十个最频繁的词对中，有九个是由两个停用词组成的， 只有一个与“the time”有关。 \n",
    "* 看看三元语法的频率是否表现出相同的行为方式。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(('the', 'time', 'traveller'), 59),\n",
       " (('the', 'time', 'machine'), 30),\n",
       " (('the', 'medical', 'man'), 24),\n",
       " (('it', 'seemed', 'to'), 16),\n",
       " (('it', 'was', 'a'), 15),\n",
       " (('here', 'and', 'there'), 15),\n",
       " (('seemed', 'to', 'me'), 14),\n",
       " (('i', 'did', 'not'), 14),\n",
       " (('i', 'saw', 'the'), 13),\n",
       " (('i', 'began', 'to'), 13)]"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trigram_tokens = [pair for pair in zip(corpus[:-2],corpus[1:-1],corpus[2:])]\n",
    "trigram_vocab = d2l.Vocab(trigram_tokens)\n",
    "trigram_vocab.token_freqs[:10]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "直观地对比三种模型中的词元频率：一元语法、二元语法和三元语法"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
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    "bigram_freqs = [freq for token,freq in bigram_vocab.token_freqs]\n",
    "trigram_freqs = [freq for token,freq in trigram_vocab.token_freqs]\n",
    "d2l.plot([freqs,bigram_freqs,trigram_freqs],xlabel='token:x',\n",
    "        ylabel='frequency:n(x)',xscale='log',yscale='log',\n",
    "        legend=['unigram','bigram','trigram'])"
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   "metadata": {},
   "source": [
    "* 首先，除了一元语法词，单词序列似乎也遵循齐普夫定律， 尽管指数$\\alpha$更小\n",
    "* 其次，词表中n元组的数量并没有那么大，这说明语言中存在相当多的结构， 这些结构给了我们应用模型的希望。 \n",
    "* 第三，很多n元组很少出现，这使得拉普拉斯平滑非常不适合语言建模。  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 读取长序列数据\n",
    "* 当序列变得太长而不能被模型一次性全部处理时， 我们可能希望拆分这样的序列方便模型读取。\n",
    "* 假设我们将使用神经网络来训练语言模型， 模型中的网络一次处理具有预定义长度 （例如n个时间步）的一个小批量序列。 现在的问题是如何随机生成一个小批量数据的特征和标签以供读取。\n",
    "* 首先，由于文本序列可以是任意长的， 例如整本《时光机器》（The Time Machine）， 于是任意长的序列可以被我们划分为具有相同时间步数的子序列。 当训练我们的神经网络时，这样的小批量子序列将被输入到模型中。 假设网络一次只处理具有n个时间步的子序列。   \n",
    "![two.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h0yywvai0uj30ez089q5j.jpg)      \n",
    "* 如果我们只选择一个偏移量， 那么用于训练网络的、所有可能的子序列的覆盖范围将是有限的。 因此，我们可以从随机偏移量开始划分序列， 以同时获得覆盖性（coverage）和随机性（randomness）。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "随机采样        \n",
    "* 在随机采样中，每个样本都是在原始的长序列上任意捕获的子序列。 \n",
    "* 在迭代过程中，来自两个相邻的、随机的、小批量中的子序列不一定在原始序列上相邻。 \n",
    "* 对于语言建模，目标是基于到目前为止我们看到的词元来预测下一个词元， 因此标签是移位了一个词元的原始序列。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "从数据中随机生成一个小批量。 \n",
    "* 在这里，参数batch_size指定了每个小批量中子序列样本的数目， 参数num_steps是每个子序列中预定义的时间步数。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "def seq_data_iter_random(corpus,batch_size,num_steps):\n",
    "    \"\"\"使用随机抽样生成一个小批量子序列\"\"\"\n",
    "    # 从随机偏移量开始对序列进行分区，随机范围包括num_steps - 1\n",
    "    corpus = corpus[random.randint(0,num_steps-1):]\n",
    "    # 减去1，是因为我们需要考虑标签\n",
    "    num_subseqs = (len(corpus) - 1) // num_steps\n",
    "    # 长度为num_steps的子序列的初始索引\n",
    "    initial_indices = list(range(0,num_subseqs*num_steps,num_steps))\n",
    "    # 在随机抽样的迭代过程中，\n",
    "    # 来自两个相邻的、随机的、小批量中的子序列不一定在原始序列上相邻\n",
    "    random.shuffle(initial_indices)\n",
    "\n",
    "    def data(pos):\n",
    "        # 返回从pos位置开始的长度为num_steps的序列\n",
    "        return corpus[pos:pos+num_steps]\n",
    "\n",
    "    num_batches = num_subseqs // batch_size\n",
    "    for i in range(0,batch_size*num_batches,batch_size):\n",
    "        # 在这里，inital_indices包含子序列的随机起始索引\n",
    "        initial_indices_per_batch = initial_indices[i:i+batch_size]\n",
    "        X = [data(j) for j in initial_indices_per_batch]\n",
    "        Y = [data(j + 1) for j in initial_indices_per_batch]\n",
    "        yield torch.tensor(X),torch.tensor(Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "下面我们生成一个从0到34的序列。 假设批量大小为2，时间步数为5，这意味着可以生成[(35-1)/5] = 6 个“特征－标签”子序列对。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X: tensor([[25, 26, 27, 28, 29],\n",
      "        [10, 11, 12, 13, 14]]) \n",
      "Y: tensor([[26, 27, 28, 29, 30],\n",
      "        [11, 12, 13, 14, 15]])\n",
      "X: tensor([[0, 1, 2, 3, 4],\n",
      "        [5, 6, 7, 8, 9]]) \n",
      "Y: tensor([[ 1,  2,  3,  4,  5],\n",
      "        [ 6,  7,  8,  9, 10]])\n",
      "X: tensor([[20, 21, 22, 23, 24],\n",
      "        [15, 16, 17, 18, 19]]) \n",
      "Y: tensor([[21, 22, 23, 24, 25],\n",
      "        [16, 17, 18, 19, 20]])\n"
     ]
    }
   ],
   "source": [
    "my_seq = list(range(35))\n",
    "for X,Y in seq_data_iter_random(my_seq,batch_size=2,num_steps=5):\n",
    "    print('X:',X,'\\nY:',Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "顺序分区    \n",
    "* 在迭代过程中，除了对原始序列可以随机抽样外， 我们还可以保证两个相邻的小批量中的子序列在原始序列上也是相邻的。 \n",
    "* 这种策略在基于小批量的迭代过程中保留了拆分的子序列的顺序，因此称为顺序分区。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [],
   "source": [
    "def seq_data_iter_sequential(corpus, batch_size, num_steps):  #@save\n",
    "    \"\"\"使用顺序分区生成一个小批量子序列\"\"\"\n",
    "    # 从随机偏移量开始划分序列\n",
    "    offset = random.randint(0, num_steps)\n",
    "    num_tokens = ((len(corpus) - offset - 1) // batch_size) * batch_size\n",
    "    Xs = torch.tensor(corpus[offset: offset + num_tokens])\n",
    "    Ys = torch.tensor(corpus[offset + 1: offset + 1 + num_tokens])\n",
    "    Xs, Ys = Xs.reshape(batch_size, -1), Ys.reshape(batch_size, -1)\n",
    "    num_batches = Xs.shape[1] // num_steps\n",
    "    for i in range(0, num_steps * num_batches, num_steps):\n",
    "        X = Xs[:, i: i + num_steps]\n",
    "        Y = Ys[:, i: i + num_steps]\n",
    "        yield X, Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 基于相同的设置，通过顺序分区读取每个小批量的子序列的特征X和标签Y。 \n",
    "* 通过将它们打印出来可以发现： 迭代期间来自两个相邻的小批量中的子序列在原始序列中确实是相邻的。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X: tensor([[ 0,  1,  2,  3,  4],\n",
      "        [17, 18, 19, 20, 21]]) \n",
      "Y: tensor([[ 1,  2,  3,  4,  5],\n",
      "        [18, 19, 20, 21, 22]])\n",
      "X: tensor([[ 5,  6,  7,  8,  9],\n",
      "        [22, 23, 24, 25, 26]]) \n",
      "Y: tensor([[ 6,  7,  8,  9, 10],\n",
      "        [23, 24, 25, 26, 27]])\n",
      "X: tensor([[10, 11, 12, 13, 14],\n",
      "        [27, 28, 29, 30, 31]]) \n",
      "Y: tensor([[11, 12, 13, 14, 15],\n",
      "        [28, 29, 30, 31, 32]])\n"
     ]
    }
   ],
   "source": [
    "for X,Y in seq_data_iter_sequential(my_seq,batch_size=2,num_steps=5):\n",
    "    print('X:',X,'\\nY:',Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "将上面的两个采样函数包装到一个类中， 以便稍后可以将其用作数据迭代器"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [],
   "source": [
    "class SeqDataLoader:\n",
    "    \"\"\"加载序列数据的迭代器\"\"\"\n",
    "    def __init__(self,batch_size,num_steps,use_random_iter,max_tokens):\n",
    "        if use_random_iter:\n",
    "            self.data_iter_fn = d2l.seq_data_iter_random\n",
    "        else:\n",
    "            self.data_iter_fn = d2l.seq_data_iter_sequential\n",
    "            self.corpus,self.vocab = d2l.load_corpus_time_machine(max_tokens)\n",
    "            self.batch_size,self.num_steps = batch_size,num_steps\n",
    "    def __iter__(self):\n",
    "        return self.data_iter_fn(self.corpus,self.batch_size,self.num_steps)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "定义了一个函数load_data_time_machine， 它同时返回数据迭代器和词表， 因此可以与其他带有load_data前缀的函数 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [],
   "source": [
    "def load_data_time_machine(batch_size, num_steps,  #@save\n",
    "                           use_random_iter=False, max_tokens=10000):\n",
    "    \"\"\"返回时光机器数据集的迭代器和词表\"\"\"\n",
    "    data_iter = SeqDataLoader(\n",
    "        batch_size, num_steps, use_random_iter, max_tokens)\n",
    "    return data_iter, data_iter.vocab"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 小结\n",
    "* 语言模型是自然语言处理的关键。\n",
    "* n元语法通过截断相关性，为处理长序列提供了一种实用的模型。\n",
    "* 长序列存在一个问题：它们很少出现或者从不出现。\n",
    "* 齐普夫定律支配着单词的分布，这个分布不仅适用于一元语法，还适用于其他元语法。\n",
    "* 通过拉普拉斯平滑法可以有效地处理结构丰富而频率不足的低频词词组。\n",
    "* 读取长序列的主要方式是随机采样和顺序分区。在迭代过程中，后者可以保证来自两个相邻的小批量中的子序列在原始序列上也是相邻的。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 循环神经网络"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 对于时间步t-(n-1)之前的单词， 如果我们想将其可能产生的影响合并到$x_{t}$上， 需要增加n，然而模型参数的数量也会随之呈指数增长， 因为词表V需要存储$|V|^n$个数字， 因此与其将$P(x_{t}|x_{t-1},...,x_{t-n+1})$模型化， 不如使用隐变量模型:$$P(x_{t}|x_{t-1},...,x_{1}) \\approx P(x_{t}|h_{t-1})$$\n",
    "$h_{t-1}$是隐状态（hidden state）， 也称为隐藏变量（hidden variable）， 它存储了到时间步t-1的序列信息。\n",
    "* 我们可以基于当前输入$x_{t}$和先前隐状态$h_{t-1}$来计算时间步处的任何时间的隐状态：$$h_{t} = f(x_{t},h_{t-1})$$ f函数，隐变量模型不是近似值。\n",
    "* 隐藏层和隐状态指的是两个截然不同的概念。 如上所述，隐藏层是在从输入到输出的路径上（以观测角度来理解）的隐藏的层， 而隐状态则是在给定步骤所做的任何事情（以技术角度来定义）的输入， 并且这些状态只能通过先前时间步的数据来计算。\n",
    "* 循环神经网络（recurrent neural networks，RNNs） 是具有隐状态的神经网络。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 无隐状态的神经网络\n",
    "* 单隐藏层的多层感知机: 设隐藏层的激活函数为$\\Phi$， 给定一个小批量样本$X∈R^ {n*d}$， 其中批量大小为n，输入维度为d， 则隐藏层的输出$H∈R^{n*h}$\n",
    "通过下式计算:$$H =\\Phi(XW_{xh}+b_{h}) $$,拥有的隐藏层权重参数为$W_{xh} ∈R^ {d*h}$， 偏置参数为$b_{h} ∈ R^{1*h}$， 以及隐藏单元的数目为h。\n",
    "* 将隐藏变量H用作输出层的输入。 输出层由下式给出    \n",
    "![one.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h0zzbavc76j30hk032gly.jpg)  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 有隐状态的循环神经网络\n",
    "* 对于n个序列样本的小批量， $X_{t}$的每一行对应于来自该序列的时间步处的一个样本。\n",
    "* 用$H_{t} ∈ R^{n×h}$表示时间步t的隐藏变量。 与多层感知机不同的是， 我们在这里保存了前一个时间步的隐藏变量$H_{t-1}$， 并引入了一个新的权重参数$W_{hh} ∈ R^{h×h}$， 来描述如何在当前时间步中使用前一个时间步的隐藏变量。\n",
    "* 当前时间步隐藏变量由当前时间步的输入 与前一个时间步的隐藏变量一起计算得出：$$H_t =\\Phi(X_{t}W_{xh}+H_{t-1}W_{hh}+b_{h}) $$\n",
    "* 从相邻时间步的隐藏变量$H_{t}$和$H_{t-1}$ 之间的关系可知， 这些变量捕获并保留了序列直到其当前时间步的历史信息， 就如当前时间步下神经网络的状态或记忆， 因此这样的隐藏变量被称为隐状态（hidden state）。\n",
    "* 由于在当前时间步中， 隐状态使用的定义与前一个时间步中使用的定义相同， 因此上述的计算是循环的（recurrent）。 于是基于循环计算的隐状态神经网络被命名为 循环神经网络（recurrent neural network）。 在循环神经网络中执行计算的层 称为循环层（recurrent layer）。\n",
    "* 对于时间步t，输出层的输出类似于多层感知机中的计算：$$O_{t} = H_{t}W_{hq} + b_{q}$$循环神经网络的参数包括隐藏层的权重$W_{xh}∈R^{d×h}$，$W_{hh}∈R^{h×h}$和偏置$b_{h} ∈ R^{1×h}$， 以及输出层的权重$W_{hq}∈R^{h×q}$和偏置$b_{q} ∈ R^{1×q}$。\n",
    "* 即使在不同的时间步，循环神经网络也总是使用这些模型参数。 因此，循环神经网络的参数开销不会随着时间步的增加而增加。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "在任意时间步t，隐状态的计算可以被视为：     \n",
    "1.拼接当前时间步t的输入$X_{t}$和前一时间步t-1的隐状态$H_{t-1}$；        \n",
    "2.将拼接的结果送入带有激活函数$\\Phi$的全连接层。 全连接层的输出是当前时间步t的隐状态$H_{t}$。       \n",
    "![three.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h1031it610j30h0091t9e.jpg)        "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们定义矩阵X、W_xh、H和W_hh， 它们的形状分别为(3,1)、(1,4)、(3,4)和(4,4)。 分别将X乘以W_xh，将H乘以W_hh， 然后将这两个乘法相加，我们得到一个形状为(3,4)的矩阵。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[-3.3203,  0.4962,  1.6860,  0.4389],\n",
       "        [ 1.9577, -2.3720,  0.7510, -1.2923],\n",
       "        [-1.0130,  1.2960,  0.5906, -0.1320]])"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import torch \n",
    "from d2l import torch as d2l\n",
    "X,W_xh = torch.normal(0,1,(3,1)),torch.normal(0,1,(1,4))\n",
    "H,W_hh = torch.normal(0,1,(3,4)),torch.normal(0,1,(4,4))\n",
    "torch.matmul(X,W_xh) + torch.matmul(H,W_hh)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "现在，我们沿列（轴1）拼接矩阵X和H， 沿行（轴0）拼接矩阵W_xh和W_hh, 再将这两个拼接的矩阵相乘。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[-3.3203,  0.4962,  1.6860,  0.4389],\n",
       "        [ 1.9577, -2.3720,  0.7510, -1.2923],\n",
       "        [-1.0130,  1.2960,  0.5906, -0.1320]])"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "torch.matmul(torch.cat((X,H),1),torch.cat((W_xh,W_hh),0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 基于循环神经网络的字符级语言模型\n",
    "* 设小批量大小为1，批量中的那个文本序列为“machine”。 为了简化后续部分的训练，我们考虑使用 字符级语言模型（character-level language model）， 将文本词元化为字符而不是单词。 下图演示了 如何通过基于字符级语言建模的循环神经网络， 使用当前的和先前的字符预测下一个字符。        \n",
    "![four.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h103gc9ugtj30gm09ewfu.jpg)     \n",
    "* 在训练过程中，我们对每个时间步的输出层的输出进行softmax操作， 然后利用交叉熵损失计算模型输出和标签之间的误差。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 困惑度（Perplexity）\n",
    "* 一个好的语言模型能够用高度准确的词元来预测我们接下来会看到什么。 \n",
    "* 如果想要压缩文本，我们可以根据当前词元集预测的下一个词元。 一个更好的语言模型应该能让我们更准确地预测下一个词元。 因此，它应该允许我们在压缩序列时花费更少的比特。 所以我们可以通过一个序列中所有的n个词元的交叉熵损失的平均值来衡量：$$\\frac{1}{n}\\sum_{t=1}^{n}-logP(x_{t}|x_{t-1},...,x_{1})$$其中P由语言模型给出，$x_{t}$是在时间步t从该序列中观察到的实际词元。\n",
    "* 困惑度的最好的理解是“下一个词元的实际选择数的调和平均数”。$$exp(-\\frac{1}{n}\\sum_{t=1}^{n}logP(x_{t}|x_{t-1},...,x_{1}))$$\n",
    "    * 在最好的情况下，模型总是完美地估计标签词元的概率为1。 在这种情况下，模型的困惑度为1。\n",
    "    * 在最坏的情况下，模型总是预测标签词元的概率为0。 在这种情况下，困惑度是正无穷大。\n",
    "    * 在基线上，该模型的预测是词表的所有可用词元上的均匀分布。 在这种情况下，困惑度等于词表中唯一词元的数量。 事实上，如果我们在没有任何压缩的情况下存储序列， 这将是我们能做的最好的编码方式。 因此，这种方式提供了一个重要的上限， 而任何实际模型都必须超越这个上限。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 小结\n",
    "* 对隐状态使用循环计算的神经网络称为循环神经网络（RNN）。\n",
    "* 循环神经网络的隐状态可以捕获直到当前时间步序列的历史信息。\n",
    "* 循环神经网络模型的参数数量不会随着时间步的增加而增加。\n",
    "* 我们可以使用循环神经网络创建字符级语言模型。\n",
    "* 我们可以使用困惑度来评价语言模型的质量。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 循环神经网络的从零开始实现\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "导入库函数和参数设置"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import math\n",
    "import torch\n",
    "from torch import nn\n",
    "from torch.nn import functional as F\n",
    "from d2l import torch as d2l\n",
    "\n",
    "batch_size,num_steps = 32,35\n",
    "train_iter,vocab = d2l.load_data_time_machine(batch_size,num_steps)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 独热编码(One-hot)\n",
    "* 我们通常将每个词元表示为更具表现力的特征向量。 最简单的表示称为独热编码\n",
    "* 将每个索引映射为相互不同的单位向量： 假设词表中不同词元的数目为（即len(vocab)）， 词元索引的范围为0到N-1。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n",
       "         0, 0, 0, 0],\n",
       "        [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n",
       "         0, 0, 0, 0]])"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F.one_hot(torch.tensor([0,2]),len(vocab))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 我们每次采样的小批量数据形状是二维张量： （批量大小，时间步数）。 one_hot函数将这样一个小批量数据转换成三维张量， 张量的最后一个维度等于词表大小（len(vocab)）。 \n",
    "* 我们经常转换输入的维度，以便获得形状为 （时间步数，批量大小，词表大小）的输出。 这将使我们能够更方便地通过最外层的维度， 一步一步地更新小批量数据的隐状态。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "torch.Size([5, 2, 28])"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = torch.arange(10).reshape((2,5))\n",
    "F.one_hot(X.T,28).shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 初始化模型参数\n",
    "* 隐藏单元数num_hiddens是一个可调的超参数。 \n",
    "* 当训练语言模型时，输入和输出来自相同的词表。 因此，它们具有相同的维度，即词表的大小。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_params(vocab_size,num_hiddens,device):\n",
    "    num_inputs = num_outputs = vocab_size\n",
    "    def normal(shape):\n",
    "        return torch.randn(size=shape,device=device) * 0.01\n",
    "    # 隐藏层参数\n",
    "    W_xh = normal((num_inputs,num_hiddens))\n",
    "    W_hh = normal((num_hiddens,num_hiddens))\n",
    "    b_h = torch.zeros(num_hiddens,device=device)\n",
    "    #输出层参数\n",
    "    W_hq = normal((num_hiddens,num_outputs))\n",
    "    b_q = torch.zeros(num_outputs,device=device)\n",
    "    # 附加梯度\n",
    "    params = [W_xh,W_hh,b_h,W_hq,b_q]\n",
    "    for param in params:\n",
    "        param.requires_grad_(True)\n",
    "    return params"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 循环神经网络模型\n",
    "为了定义循环神经网络模型， 我们首先需要一个init_rnn_state函数在初始化时返回隐状态。 这个函数的返回是一个张量，张量全用0填充， 形状为（批量大小，隐藏单元数）。 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [],
   "source": [
    "def init_rnn_state(batch_size,num_hidens,device):\n",
    "    return (torch.zeros((batch_size,num_hidens),device=device),)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 下面的rnn函数定义了如何在一个时间步内计算隐状态和输出。 循环神经网络模型通过inputs最外层的维度实现循环， 以便逐时间步更新小批量数据的隐状态H。 \n",
    "* 此外，这里tanh使用函数作为激活函数。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [],
   "source": [
    "def rnn(inputs,state,params):\n",
    "    # inputs的形状，(时间步数量，批量大小，词表大小)\n",
    "    W_xh,W_hh,b_h,W_hq,b_q = params\n",
    "    H, = state\n",
    "    outputs = []\n",
    "    # X的形状：(批量大小，词表大小)\n",
    "    for X in inputs:\n",
    "        H = torch.tanh(torch.mm(X,W_xh) + torch.mm(H,W_hh) + b_h)\n",
    "        Y = torch.mm(H,W_hq) + b_q\n",
    "        outputs.append(Y)\n",
    "    return torch.cat(outputs,dim=0),(H,)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "定义了所有需要的函数之后，接下来我们创建一个类来包装这些函数， 并存储从零开始实现的循环神经网络模型的参数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [],
   "source": [
    "class RNNModelScratch:\n",
    "    \"\"\"从零实现的循环神经网络\"\"\"\n",
    "    def __init__(self,vocab_size,num_hiddens,device,get_params,init_state,forward_fn):\n",
    "        self.vocab_size,self.num_hiddens = vocab_size,num_hiddens\n",
    "        self.params = get_params(vocab_size,num_hiddens,device)\n",
    "        self.init_state,self.forward_fn = init_state,forward_fn\n",
    "    \n",
    "    def __call__(self,X,state):\n",
    "        X = F.one_hot(X.T,self.vocab_size).type(torch.float32)\n",
    "        return self.forward_fn(X,state,self.params)\n",
    "    \n",
    "    def begin_state(self,batch_size,device):\n",
    "        return self.init_state(batch_size,self.num_hiddens,device)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "让我们检查输出是否具有正确的形状"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(torch.Size([10, 28]), 1, torch.Size([2, 512]))"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "num_hiddens = 512\n",
    "net = RNNModelScratch(len(vocab),num_hiddens,d2l.try_gpu(),get_params,init_rnn_state,rnn)\n",
    "state = net.begin_state(X.shape[0],d2l.try_gpu())\n",
    "Y,new_state = net(X.to(d2l.try_gpu()),state)\n",
    "Y.shape,len(new_state),new_state[0].shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 预测\n",
    "* 让我们首先定义预测函数来生成prefix之后的新字符， 其中的prefix是一个用户提供的包含多个字符的字符串。 \n",
    "* 在循环遍历prefix中的开始字符时， 我们不断地将隐状态传递到下一个时间步，但是不生成任何输出。 这被称为预热（warm-up）期， 因为在此期间模型会自我更新（例如，更新隐状态）， 但不会进行预测。 \n",
    "* 预热期结束后，隐状态的值通常比刚开始的初始值更适合预测， 从而预测字符并输出它们。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [],
   "source": [
    "def predict_ch8(prefix,num_preds,net,vocab,device):\n",
    "    \"\"\"在prefix后面生成新字符\"\"\"\n",
    "    state = net.begin_state(batch_size=1,device=device)\n",
    "    outputs = [vocab[prefix[0]]]\n",
    "    get_input = lambda: torch.tensor([outputs[-1]],device=device).reshape((1,1))\n",
    "    for y in prefix[1:]: #预热期\n",
    "        _,state = net(get_input(),state)\n",
    "        outputs.append(vocab[y])\n",
    "    for _ in range(num_preds): #预测num_preds步\n",
    "        y,state = net(get_input(),state)\n",
    "        outputs.append(int(y.argmax(dim=1).reshape(1)))\n",
    "    return ''.join([vocab.idx_to_token[i] for i in outputs])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "将前缀指定为time traveller， 并基于这个前缀生成10个后续字符。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'time traveller qjvfpolmy'"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "predict_ch8('time traveller ',9,net,vocab,d2l.try_gpu())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 梯度裁剪\n",
    "* 对于长度为T的序列，我们在迭代中计算这个时间步T上的梯度， 将会在反向传播过程中产生长度为O(T)的矩阵乘法链。\n",
    "* 当T较大时，它可能导致数值不稳定， 例如可能导致梯度爆炸或梯度消失。 因此，循环神经网络模型往往需要额外的方式来支持稳定训练。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果我们进一步假设目标函数f表现良好， 即函数在常数下是利普希茨连续的（Lipschitz continuous）。 也就是说，对于任意x和y我们有：       \n",
    "![six.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h1098k4pskj30he0613zn.jpg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "定义一个函数来裁剪模型的梯度， 模型是从零开始实现的模型或由高级API构建的模型。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [],
   "source": [
    "def grad_clipping(net,theta):\n",
    "    \"\"\"裁剪梯度\"\"\"\n",
    "    if isinstance(net,nn.Module):\n",
    "        params = [p for p in net.parameters() if p.requires_grad]\n",
    "    else:\n",
    "        params = net.params\n",
    "    norm = torch.sqrt(sum(torch.sum((p.grad ** 2)) for p in params))\n",
    "    if norm > theta:\n",
    "        for param in params:\n",
    "            param.grad[:] *= theta /norm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 训练\n",
    "在训练模型之前，让我们定义一个函数在一个迭代周期内训练模型。 它与之前训练模型的方式有三个不同之处：\n",
    "* 序列数据的不同采样方法（随机采样和顺序分区）将导致隐状态初始化的差异。\n",
    "* 我们在更新模型参数之前裁剪梯度。 这样的操作的目的是：即使训练过程中某个点上发生了梯度爆炸，也能保证模型不会发散。\n",
    "* 我们用困惑度来评价模型"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 当使用顺序分区时， 我们只在每个迭代周期的开始位置初始化隐状态。 由于下一个小批量数据中的第i个子序列样本 与当前第i个子序列样本相邻， 因此当前小批量数据最后一个样本的隐状态， 将用于初始化下一个小批量数据第一个样本的隐状态。\n",
    "* 在任何一点隐状态的计算， 都依赖于同一迭代周期中前面所有的小批量数据， 这使得梯度计算变得复杂。为了降低计算量，在处理任何一个小批量数据之前， 我们先分离梯度，使得隐状态的梯度计算总是限制在一个小批量数据的时间步内。\n",
    "* 当使用随机抽样时，因为每个样本都是在一个随机位置抽样的， 因此需要为每个迭代周期重新初始化隐状态。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [],
   "source": [
    "def train_epoch_ch8(net,train_iter,loss,updater,device,use_random_iter):\n",
    "    \"\"\"训练网络一个迭代周期\"\"\"\n",
    "    state,timer = None,d2l.Timer()\n",
    "    metric = d2l.Accumulator(2) # 训练损失之和，词元数量\n",
    "    for X,Y in train_iter:\n",
    "        if state is None or use_random_iter:\n",
    "            # 在第一次迭代或使用随机抽样时初始化state\n",
    "            state = net.begin_state(batch_size=X.shape[0],device=device)\n",
    "        else:\n",
    "            if isinstance(net,nn.Module) and not isinstance(state,tuple):\n",
    "                # state对于nn.GRU是一个张量\n",
    "                state.detach_()\n",
    "            else:\n",
    "                # state对于nn.Lstm或对于我们从零开始实现的模型是张量\n",
    "                for s in state:\n",
    "                    s.detach_()\n",
    "        y = Y.T.reshape(-1)\n",
    "        X,y = X.to(device),y.to(device)\n",
    "        y_hat,state = net(X,state)\n",
    "        l = loss(y_hat,y.long()).mean()\n",
    "        if isinstance(updater,torch.optim.Optimizer):\n",
    "            updater.zero_grad()\n",
    "            l.backward()\n",
    "            grad_clipping(net,1)\n",
    "            updater.step()\n",
    "        else:\n",
    "            l.backward()\n",
    "            grad_clipping(net,1)\n",
    "            # 因为已经调用mean函数\n",
    "            updater(batch_size=1)\n",
    "        metric.add(l*y.numel(),y.numel())\n",
    "    return math.exp(metric[0]/metric[1]),metric[1]/timer.stop()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "循环神经网络模型的训练函数既支持从零开始实现， 也可以使用高级API来实现"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [],
   "source": [
    "def train_ch8(net,train_iter,vocab,lr,num_epochs,device,use_random_iter=False):\n",
    "    \"\"\"训练模型\"\"\"\n",
    "    loss = nn.CrossEntropyLoss()\n",
    "    animator = d2l.Animator(xlabel='epoch',ylabel='perplexity',legend=['train'],xlim = [10,num_epochs])\n",
    "    # 初始化\n",
    "    if isinstance(net,nn.Module):\n",
    "        updater = torch.optim.SGD(net.parameters(),lr)\n",
    "    else:\n",
    "        updater = lambda batch_size:d2l.sgd(net.params,lr,batch_size)\n",
    "    predict = lambda prefix:predict_ch8(prefix,50,net,vocab,device)\n",
    "    # 训练和预测\n",
    "    for epoch in range(num_epochs):\n",
    "        ppl,speed =train_epoch_ch8(\n",
    "            net,train_iter,loss,updater,device,use_random_iter\n",
    "        )\n",
    "        if (epoch+1) % 10 == 0:\n",
    "            print(predict('time traveller'))\n",
    "            animator.add(epoch+1,[ppl])\n",
    "    print(f'困惑度{ppl:.1f},{speed:.1f} 词元/秒 {str(device)}')\n",
    "    print(predict('time traveller'))\n",
    "    print(predict('traveller'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们训练循环神经网络模型。 因为我们在数据集中只使用了10000个词元， 所以模型需要更多的迭代周期来更好地收敛。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "困惑度1.0,51772.5 词元/秒 cuda:0\n",
      "time travelleryou can show black is white by argument said filby\n",
      "traveller with a slight accession ofcheerfulness really thi\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\r\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\r\n",
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   "source": [
    "num_epochs,lr = 500,1\n",
    "train_ch8(net,train_iter,vocab,lr,num_epochs,d2l.try_gpu())"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "使用随机抽样方法的结果"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "困惑度1.4,48826.7 词元/秒 cuda:0\n",
      "time travellerit s against reason said filbywan a sube thave beg\n",
      "travellerit s against reason said filbywan a sube thave beg\n"
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   ],
   "source": [
    "net = RNNModelScratch(len(vocab), num_hiddens, d2l.try_gpu(), get_params,\n",
    "                      init_rnn_state, rnn)\n",
    "train_ch8(net, train_iter, vocab, lr, num_epochs, d2l.try_gpu(),\n",
    "          use_random_iter=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 小结\n",
    "* 我们可以训练一个基于循环神经网络的字符级语言模型，根据用户提供的文本的前缀生成后续文本。\n",
    "* 一个简单的循环神经网络语言模型包括输入编码、循环神经网络模型和输出生成。\n",
    "* 循环神经网络模型在训练以前需要初始化状态，不过随机抽样和顺序划分使用初始化方法不同。\n",
    "* 当使用顺序划分时，我们需要分离梯度以减少计算量。\n",
    "* 在进行任何预测之前，模型通过预热期进行自我更新（例如，获得比初始值更好的隐状态）。\n",
    "* 梯度裁剪可以防止梯度爆炸，但不能应对梯度消失。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 循环神经网络的简洁实现"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "导入库函数\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import torch \n",
    "from torch import nn\n",
    "from torch.nn import functional as F\n",
    "from d2l import torch as d2l\n",
    "\n",
    "batch_size,num_steps = 32,35\n",
    "train_iter,vocab = d2l.load_data_time_machine(batch_size,num_steps)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 定义模型\n",
    "构造一个具有256个隐藏单元的单隐藏层的循环神经网络层rnn_layer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "num_hiddens = 256\n",
    "rnn_layer = nn.RNN(len(vocab),num_hiddens)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "使用张量来初始化隐状态，它的形状是（隐藏层数，批量大小，隐藏单元数）"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "torch.Size([1, 32, 256])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "state = torch.zeros((1,batch_size,num_hiddens))\n",
    "state.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 通过一个隐状态和一个输入，我们就可以用更新后的隐状态计算输出。 \n",
    "* 需要强调的是，rnn_layer的“输出”（Y）不涉及输出层的计算： 它是指每个时间步的隐状态，这些隐状态可以用作后续输出层的输入。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(torch.Size([35, 32, 256]), torch.Size([1, 32, 256]))"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = torch.rand(size=(num_steps,batch_size,len(vocab)))\n",
    "Y,state_new = rnn_layer(X,state)\n",
    "Y.shape,state_new.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们为一个完整的循环神经网络模型定义了一个RNNModel类。 注意，rnn_layer只包含隐藏的循环层，我们还需要创建一个单独的输出层。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "class RNNModel(nn.Module):\n",
    "    \"\"\"循环神经网络模型\"\"\"\n",
    "    def __init__(self,rnn_layer,vocab_size,**kwargs):\n",
    "        super(RNNModel,self).__init__(**kwargs)\n",
    "        self.rnn = rnn_layer\n",
    "        self.vocab_size = vocab_size\n",
    "        self.num_hiddens = self.rnn.hidden_size\n",
    "        # 如果RNN是双向的，num_directions应该是2，否则应该是1\n",
    "        if not self.rnn.bidirectional:\n",
    "            self.num_directions = 1\n",
    "            self.linear = nn.Linear(self.num_hiddens,self.vocab_size)\n",
    "        else:\n",
    "            self.num_directions = 2\n",
    "            self.linear = nn.Linear(self.num_hiddens * 2,self.vocab_size)\n",
    "    def forward(self,inputs,state):\n",
    "        X = F.one_hot(inputs.T.long(),self.vocab_size)\n",
    "        X = X.to(torch.float32)\n",
    "        Y,state = self.rnn(X,state)\n",
    "        # 全连接层首先将Y的形状改为(时间步数*批量大小，隐藏单元数)\n",
    "        # 它的输出形状是(时间步数*批量大小，词表大小)\n",
    "        output = self.linear(Y.reshape((-1,Y.shape[-1])))\n",
    "        return output,state\n",
    "\n",
    "    def begin_state(self,device,batch_size=1):\n",
    "        if not isinstance(self.rnn,nn.LSTM):\n",
    "            # nn.GRU以张量作为隐状态\n",
    "            return torch.zeros((self.num_directions * self.rnn.num_layers,\n",
    "                                batch_size,self.num_hiddens),\n",
    "                                device=device)\n",
    "        else: \n",
    "            # nn.LSTM以元组作为隐状态\n",
    "            return (torch.zeros((\n",
    "                    self.num_directions * self.rnn.num_layers,\n",
    "                    batch_size,self.num_hiddens),device=device),\n",
    "                    torch.zeros((\n",
    "                        self.num_directions*self.rnn.num_layers,\n",
    "                        batch_size,self.num_hiddens),device=device)\n",
    "                    )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 训练与预测\n",
    "在训练模型之前，让我们基于一个具有随机权重的模型进行预测。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'time travellervpvvvvvvvv'"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "device = d2l.try_gpu()\n",
    "net = RNNModel(rnn_layer,vocab_size=len(vocab))\n",
    "net = net.to(device)\n",
    "d2l.predict_ch8('time traveller',10,net,vocab,device)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "定义的超参数调用train_ch8，并且使用高级API训练模型"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "perplexity 1.3, 163346.5 tokens/sec on cuda:0\n",
      "time traveller ffourd ioneid fowr way expurny f fouridid tha fic\n",
      "traveller hild and and do s amatyou hant wer at als allonet\n"
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   ],
   "source": [
    "num_epochs,lr = 500,1\n",
    "d2l.train_ch8(net,train_iter,vocab,lr,num_epochs,device)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 小结\n",
    "* 深度学习框架的高级API提供了循环神经网络层的实现。\n",
    "* 高级API的循环神经网络层返回一个输出和一个更新后的隐状态，我们还需要计算整个模型的输出层。\n",
    "* 相比从零开始实现的循环神经网络，使用高级API实现可以加速训练。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 通过时间反向传播\n",
    "* 通过时间反向传播（backpropagation through time，BPTT，实际上是循环神经网络中反向传播技术的一个特定应用。\n",
    "* 它要求我们将循环神经网络的计算图一次展开一个时间步， 以获得模型变量和参数之间的依赖关系。 然后，基于链式法则，应用反向传播来计算和存储梯度。 由于序列可能相当长，因此依赖关系也可能相当长。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 循环神经网络的梯度分析\n",
    "* 我们分别使用$w_{h}$和$w_{o}$来表示隐藏层和输出层的权重。 每个时间步的隐状态和输出可以写为：$$h_{t} = f(x_{t},h_{t-1},w_{h})$$ $$o_{t} = g(h_{t},w_{o})$$其中f和g分别是隐藏层和输出层的变换。\n",
    "*  前向传播相当简单，一次一个时间步的遍历三元组$(x_{t},h_{t},o_{t})$， 然后通过一个目标函数在所有T个时间步内评估输出$o_{t}$和对应的标签$y_{t}$之间的差异：$$L(x_{1},...,x_{T},y_{1},...,y_{T},w_{h},w_{o}) = \\frac{1}{T} \\sum_{t=1}^{T}l(y_{t},o_{t}) $$\n",
    "* 按照链式法则：$$\\frac{ \\partial L}{ \\partial w_{h}} = \\frac{1}{T} \\sum_{t=1}^{T}  \\frac {\\partial l(y_{t},o_{t})}{ \\partial w_{h}} = \\frac{1}{T} \\sum_{t=1}^{T}  \\frac {\\partial l(y_{t},o_{t})}{ \\partial o_{t}}  \\frac {\\partial g(h_{t},w_{o})}{ \\partial h_{t}}  \\frac {\\partial h_{t}}{ \\partial w_{h}}$$,第三项计算较为复杂，推导如下       \n",
    "![one.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h119ygy19yj30p30e4q5c.jpg)      "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "截断时间步      \n",
    "* 我们可以在$\\tau$步后截断中的求和计算。这会带来真实梯度的近似， 只需将求和终止为$\\partial h_{t-\\tau} / \\partial w_{h} $。在实践中，这种方式工作得很好。 它通常被称为截断的通过时间反向传播。这样做导致该模型主要侧重于短期影响，而不是长期影响。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "随机截断\n",
    "* 我们可以用一个随机变量替换$\\partial h_{t} / \\partial w_{h} $， 该随机变量在预期中是正确的，但是会截断序列。梯度$\\partial h_{t-\\tau} / \\partial w_{h} $得到：      \n",
    "![two.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h11a7d5obpj30be01s0sq.jpg)      "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "比较策略：      \n",
    "![three.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h11a9eolx6j30g805s0t4.jpg)\n",
    "* 第一行采用随机截断，方法是将文本划分为不同长度的片断。\n",
    "* 第二行采用常规截断，方法是将文本分解为相同长度的子序列。 这也是我们在循环神经网络实验中一直在做的。\n",
    "* 第三行采用通过时间的完全反向传播，结果是产生了在计算上不可行的表达式。\n",
    "\n",
    "虽然随机截断在理论上具有吸引力， 但很可能是由于多种因素在实践中并不比常规截断更好。 首先，在对过去若干个时间步经过反向传播后， 观测结果足以捕获实际的依赖关系。 其次，增加的方差抵消了时间步数越多梯度越精确的事实。 第三，我们真正想要的是只有短范围交互的模型。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 通过时间反向传播的细节\n",
    "![four.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h11acew76fj30gj090gne.jpg)     \n",
    "模型参数是$W_{hx}$、$W_{hh}$和W_{qh}。通常，训练该模型需要对这些参数进行梯度计算：$\\partial L / \\partial W_{hx} $、$\\partial L / \\partial W_{hh} $和$\\partial L / \\partial W_{qh}$。        \n",
    "![1.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h11aiplobcj306201jglh.jpg)    \n",
    "![2.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h11ajczud2j30br020dfw.jpg)    \n",
    "![3.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h11ajlxmxuj30an01oq2x.jpg)    \n",
    "![4.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h11ajsb9hgj30is01v0sw.jpg)    \n",
    "![5.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h11ajzpr4nj309d029q2v.jpg)    \n",
    "![6.png](https://image.baidu.com/search/down?url=https://tva1.sinaimg.cn/large/005T39qaly1h11ak6tkxbj30cl03qgm0.jpg)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 小结\n",
    "* “通过时间反向传播”仅仅适用于反向传播在具有隐状态的序列模型。\n",
    "* 截断是计算方便性和数值稳定性的需要。截断包括：规则截断和随机截断。\n",
    "* 矩阵的高次幂可能导致神经网络特征值的发散或消失，将以梯度爆炸或梯度消失的形式表现。\n",
    "* 为了计算的效率，“通过时间反向传播”在计算期间会缓存中间值。"
   ]
  }
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